A parallel beam of light is incident from air at an angle `alpha` on the side of right angled triangular prism of refractive index `mu = sqrt(2)`. Light undergoes total internal reflection in the prism at the face `PR` when `alpha` has a minimum value of `45^@`. The angle `theta` of the prism is.
.

Options

A. `15^(@)`
B. `22.5^(@)`
C. `30^(@)`
D. `45^(@)`

Answer

A

Explanation

Critical angle for this material in air can be written as:
`theta_(c) = sin^(-1) = 45^(@)`.
It is given that when a is set at a minimum of `45^(@)` then light ray is just able to suffer total internal reflection at the face PR. So we can understand that when `alpha = 45^(@)` then angle `beta` shown in figure is equal to critical angle which is also equal to `45^(@)`.
Now we can write the following condition for angles:
`90^(@)+beta=theta+(90^(@)+r)`
`rArr" "45^(@)=theta+r`
`rArr" "r=45^(@)-theta`
Now we can use Snell.s law at the point where ray enters the prism.
`mu sin theta="const"`
`1xxsin alpha=sqrt2xxsinr`
`rArr" "sin45^(@)=sqrt2sin(45-theta)`
`rArr" "sin(45-theta)=1//2`
`rArr" "45-theta=30`
`rArr" "theta=15^(@)`
Hence, option (a) is correct.